200      IX   ANALYTIC FUNCTIONS
For if \\cvbv\\ < A for any veNp, it follows from the definition of the
norm in Kp that if \zt\ < rt < \bt\ (1 < /</>), we have ||cvzv|| < Aq\ with
q = (qi,...,qp), qi^rj\bt\<l. It follows from (5.5.3) that the family
(qv)vem of positive numbers is absolutely summable, hence the result by
(5.3.1).
(9.1.3)   Under the assumptions 0/(9.1.2), the sum of the power series (cvzv) is
continuous in the open poly disk of center 0 and radii \bt\.
As every point of that polydisk is interior to a closed polydisk of
radii rl < \bt\9 the result follows from (7.2.1).
Let q be any integer such that l^q^p; for any v = (rcl5 ...,«),
write / = («!,..., nq), v" = (fl?+!,..., «p); consider Kp as identified to
the product Kq x Kp~«, and for z = (zi ..., zp) e Kp, write z' = (zl9..., Zq),
z" = (zq+1,..., zp). With these notations:
(9.1.4) Suppose the power series (cvzv) is absolutely summable in the polydisk
P of radii rt and center 0 in Kp. Then, for any v" e Np~~q the series (c(v,t v,}z/v')
is absolutely summable in the poly disk P', projection of P on Kq; let #v»(z')
be its sum. Then, for any zr e P', the power series (gv»(z')z"v") is absolutely
summable in the polydisk P", projection ofP on YJ>~~q, and its sum is equal to
the sum of the series (cvzv).
As zv = z/vV/v", the fact that each of the series (<v,v)z'v'z"v") (v" fixed)
is absolutely summable, and that ]£ #v,,(z')z"v" = ]T cvzv, follows from (5.3.5)
v"	v
and from the associativity theorem (5.3.6) for absolutely summable families.
If we take z" e P" such that zi ^ 0 for q + 1 < / < /?, the absolute summability
of (C(V<t V")z/v') follows.
(9.1.5)      ("Principle of isolated zeros")   Suppose (cnzn) is a power series
in one variable which  converges in  an  open   ball P  of radius  r, and
00
let /(z) = ^] cnzn. Then, unless all  the cn   are 0,   there  is r' <r such
n=0
that for 0< \z\<r',f(z)*Q.
Suppose h is the smallest integer such that ck^Q; then we can write
/(z) = zh (ch + ch+1z + - • • + ch+mzm + • • •) and the series (ch+mzm) converges
in P; if g(z) = ch + ch+1z -f • • • + ch+mzm + ..., g is continuous in P by
(9.1.3) and as 0(0) = ch ^ 0, there is r' > 0 such that g(z) ^ 0 for |z| < r';
hence the result.